Properties

Label 2.625.adt_fio
Base field $\F_{5^{4}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 49 x + 625 x^{2} )( 1 - 48 x + 625 x^{2} )$
  $1 - 97 x + 3602 x^{2} - 60625 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0637685608585$, $\pm0.0903344706017$
Angle rank:  $2$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $333506$ $151728554700$ $59593325970144968$ $23282931123531240192000$ $9094945732251198388598540066$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $529$ $388421$ $244094260$ $152587017409$ $95367418161409$ $59604644663623586$ $37252902987502815505$ $23283064365593187091969$ $14551915228374957999597364$ $9094947017729544342083496101$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5^{4}}$
The isogeny class factors as 1.625.abx $\times$ 1.625.abw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.625.ab_abqk$2$(not in LMFDB)
2.625.b_abqk$2$(not in LMFDB)
2.625.dt_fio$2$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.625.ab_abqk$2$(not in LMFDB)
2.625.b_abqk$2$(not in LMFDB)
2.625.dt_fio$2$(not in LMFDB)
2.625.acl_cwm$4$(not in LMFDB)
2.625.abj_vs$4$(not in LMFDB)
2.625.bj_vs$4$(not in LMFDB)
2.625.cl_cwm$4$(not in LMFDB)